Calculus Examples

$\frac{lim_{x \to 4^{+}} | x - 4 |}{- 8 + 2 x}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit inside the absolute value signs.

$\frac{| lim_{x \to 4^{+}} x - 4 |}{- 8 + 2 x}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{| lim_{x \to 4^{+}} x - lim_{x \to 4^{+}} 4 |}{- 8 + 2 x}$

Step 1.3

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$\frac{| lim_{x \to 4^{+}} x - 1 \cdot 4 |}{- 8 + 2 x}$

Step 2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{| 4 - 1 \cdot 4 |}{- 8 + 2 x}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $4$ .

$\frac{| 4 - 4 |}{- 8 + 2 x}$

Step 3.1.2

Subtract $4$ from $4$ .

$\frac{| 0 |}{- 8 + 2 x}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$\frac{0}{- 8 + 2 x}$

Step 3.2

Factor $2$ out of $- 8 + 2 x$ .

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Step 3.2.1

Factor $2$ out of $- 8$ .

$\frac{0}{2 \cdot - 4 + 2 x}$

Step 3.2.2

Factor $2$ out of $2 \cdot - 4 + 2 x$ .

$\frac{0}{2 (- 4 + x)}$

Step 3.3

Cancel the common factor of $0$ and $2$ .

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Step 3.3.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 (- 4 + x)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Cancel the common factor.

$\frac{2 \cdot 0}{2 (- 4 + x)}$

Step 3.3.2.2

Rewrite the expression.

$\frac{0}{- 4 + x}$

Step 3.4

Divide $0$ by $- 4 + x$ .

$0$